Given that $x^2 + y^2 = 14x + 6y + 6,$ find the largest possible value of $3x + 4y.$
Let $z = 3x + 4y.$  Then $y = \frac{z - 3x}{4}.$  Substituting into $x^2 + y^2 = 14x + 6y + 6,$ we get
\[x^2 + \left( \frac{z - 3x}{4} \right)^2 = 14x + 6 \cdot \frac{z - 3x}{4} + 6.\]This simplifies to
\[25x^2 - 6xz + z^2 - 152x - 24z - 96 = 0.\]Writing this as a quadratic in $x,$ we get
\[25x^2 - (6z + 152) x + z^2 - 24z - 96 = 0.\]This quadratic has real roots, so its discriminant is nonnnegative.  This gives us
\[(6z + 152)^2 - 4 \cdot 25 \cdot (z^2 - 24z - 96) \ge 0.\]This simplifies to $-64z^2 + 4224z + 32704 \ge 0,$ which factors as $-64(z + 7)(z - 73) \ge 0.$  Therefore, $z \le 73.$

Equality occurs when $x = \frac{59}{5}$ and $y = \frac{47}{5},$ so the maximum value is $\boxed{73}.$